Number Sense Archives - Page 6 of 7 - Mathematics for Teaching

Teaching positive and negative numbers

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A popular approach for teaching numbers is to use it to describe a property of an object or a set of object. For example, numbers are used to describe the amount or quantity of fruits in a basket.

In introducing integers, teachers and textbooks presents integers as a set of numbers that can be used to describe both the quantity and quality of an object or idea. Contexts involving opposites are very popular situations to show the uses and importance of positive and negative numbers and the meaning of its symbols. For example, a teacher can tell the class that +5 represents going 5 floors up and -5 represents going five floors down from an initial position.

Mathematics is a language and a way of thinking and should therefore be experienced by students as such. As a language, math is presented as having its own set of symbols and “grammar” much like our spoken and written languages that we use to describe a thing, an experience or an idea.But apart from being a language, mathematics is also a way of thinking. The only way for students to learn how to think is for them to engage them in it! Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking as well.

Sort the following situations according to some categories

3^o below zero
52 m below sea level
$1000 net gain
$5000 withdrawal from ATM machine
$1000 deposit in savings account
3 kg weight loss
2 kg weight gain
80 m above sea level
37^o above zero
$2000 net loss

The task may seem like an ordinary sorting task but notice that the categories are not given. Students have to make their own way of grouping the situations. They can only do this after analyzing each situation, noting commonalities and differences.

Possible solutions:

1. Distance vs money (some students may consider the reading the thermometer under distance since its about the “length” of mercury from the “base”)

2. Based on type of quantities: amount of money, temperature, mass, length

3. Based on contrasting sense: weight gain vs weight loss, above zero vs below zero, etc.

The last solution is what you want. With very little help you can guide students to come-up with the solution below.

Of course, one may wonder why make the students go through all these. Why not just tell them? Why not give the categories? Well, mathematics is not in the curriculum because we want students to just learn mathematics. More importantly, we want our students to think critically and creatively hence we need to give them learning experiences that develops good thinking habits. Mathematics is a very good context for learning these.

Here are my other posts about integers:

Misconceptions, Number Sense

From whole numbers to integers – so many things to “unlearn”

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A lot of online resources on integers are about operations on integers especially addition and subtraction. Most of these resources show visual representations of integer operations. These representations are almost always in the form of jumping bunnies, kitties, frogs, … practically anything that can or cannot jump are made to jump on the number line. Sometimes I wonder where and when in their math life will the students ever encounter or use jumping on the number line again. If you want to know why I think number line might not work for teaching operations, click link – Subtracting integers using number line – why it doesn’t help the learning.

Of course there may be other culprits apart from rote learning and the numberline model. Maybe there are other things that blocks students’ understanding of integers especially doing operations with them.

Before integers, students’ life with numbers had been all about whole numbers and some friendly fractions and decimals. So it is not surprising that they would have made some generalizations related to whole numbers with or without teachers help. I pray of course that teachers will have no hand in arriving at these generalizations and that if indeed students will come to these conclusions, it should be by the natural course of things. Here are some dangerous generalizations.

generalization about whole numbers — over-generalizations about whole numbers

These generalizations are very difficult to unlearn (accommodate according to Piaget) because based on students experiences they all work and are all true. Now, here comes integers turning all of these upside down, creating cognitive conflict. In the set of integers,

when a number is added to another number it could get smaller (5 + -3 gives 2; 2 is smaller than 5)
the sum of any two numbers can be smaller than both of the addends (-3 + -2 gives -5; -5 is smaller than -3 and -2)
when a number is taken a way from another number, it could get bigger (3 – -2 = 5, 3 just got bigger by 2)
you can get an answer for taking away a bigger number from a smaller number (3 – 5 = -2)
when a number is multiplied by another number, it could get smaller (-3 x 2 = -5)
when a number is divided by another number, it could get bigger (-15/-3 = 5)

On top of these, mathematics is taught as something that gives absolute result. So how come things change?

You may be interested to read my article on Math War over Multiplication. It’s also about overgeneralization.

Feel free to share your thoughts about these.

Elementary School Math, Number Sense

Math War over Multiplication

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The post It Ain’t No Repeated Addition by Devlin launched a math war over the definition of multiplication. Here’s an excerpt from that post:

“Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.”

Multiplication is repeated addition is definitely not correct. Counterexample: 1/2 x 1/4. Try also doing it with integers like -5 x -4 (not that you need two counterexamples to reject a statement). But is it correct to say that in the set of whole numbers multiplication is repeated addition? I think not. You can get the result by repeated addition for this set of number yes, but that does not make repeated addition a definition of multiplication. An operation is not defined by the strategy of getting its result.

But, should teachers in the grades stop telling pupils that multiplication is repeated addition? YES! In fact, they should refrain from telling pupils any rule at all. The pupils are perfectly capable of figuring things like these by themselves given the right task/activity and good facilitation by the teacher.

And let us suppose that students get this conception that multiplication is repeated addition, is there really a problem? Their world revolve around whole numbers so it’s only logical that this will be their understanding of it. Generalizing is a natural human tendency. Something must be wrong if they will not make this connection between multiplication and addition.

What is wrong with “undoing” later? Mathematics is man-made and there’s also a lot of trial and error part in its development. That is why “undoing” and rejection by counterexample are legitimate processes . And, isn’t ‘undoing’ part of teaching? Good teachers are those who can find out or know what they should be ‘undoing’ when they teach mathematics. ‘Multiplication is repeated addition’ is only one of many ‘over-generalizations’ pupils will make that teachers need to carefully undo later. There’s “when you multiply, you make it bigger”, or “the sum of two numbers is always bigger than any of the two you added”, etc. One way to prevent an over-generalization is to offer a counterexample. But where will you get that counterexample when their math still revolves around the world of whole numbers!

As teachers, don’t we all love that part of teaching where we challenge students’ assumptions? I’m not saying that we should deliberately lead pupils to over-generalizations so we have something to undo later. For example, we don’t lead them to “division is repeated subtraction”? Most of the time oversimplifying mathematics is not a good idea.

Click link to know what others say about multiplication is not repeated addition.

Fractal as multiplication model

Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

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Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers — Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers — Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions. Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

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